[mdbfifgm] a raft of flotsam and jetsam tenuously lashed together by seaweed and hope

antonym of "(running a) tight ship".

(cf the game "Raft")

[umxqnhxg] Schroedinger's cat of 9 lives

in the thought experiment of Schroedinger's cat, after the murder mechanism is triggered by half a photon, the box contains a cyanide capsule in a quantum superposition of unbroken and broken, and a cat in a quantum superposition of alive and dead.  it is NOT the case that the cat is either alive or dead but it's unknown which because the box is closed; rather, the cat truly is in the classically incomprehensible state of being simultaneously alive and dead.  (other experiments like quantum double slit support this.)

after the box is opened, the quantum state randomly "collapses" (Copenhagen interpretation) into precisely one of the two possibilities:

1. cyanide capsule unbroken and cat alive
2. cyanide capsule broken and cat dead

...which would be true if Schroedinger had chosen anything but a cat as his lab animal.  (why not a lab rat like any normal scientist?)

because cat, we must add the possibility of cyanide released but cat still alive (with its internal lives counter decreased by one).

also the small possibility of cyanide unreleased but cat dead of natural causes (perhaps curiosity) during the (presumably) brief experiment.

previously, superluminal action at a distance between two cats.  that experiment could end with both cats alive (one with its internal lives counter decreased by one).

[kxdvujsv] N queens of two colors (peaceable queens)

Q/2 white queens, Q/2 black.  how many can fit on a chessboard (i.e., maximize Q) with no queen attacking one of the opposite color?  queens of the same color may attack (actually defend) each other (but don't have to).

I'm sure this problem has been proposed before because it makes sense for regular chess. but it is hard to search for it, dominated by results of the traditional N queens problem all of the same color, (or, to be consistent with actual chess, all of different colors, because queens of the same color defending each other is not something to avoid in chess).

update: OEIS A250000, found by luck from Pictures from the OEIS, calls the problem Peaceable Queens.  the problem seems young, much younger than the chess queen itself, with the oldest reference 1977, and the OEIS sequence itself created in 2014.  (Sloane himself arranged for it to get a nice OEIS A-number because he likes the problem, with lower bounds findable by simulated annealing and upper bounds findable by integer programming.)  but certainly chess players had considered the problem earlier?

the maximum for the 8x8 board is 9 queens of each color, which coincidentally corresponds to the original queens plus all pawns promoted to queens.  how many maximal solutions are there for the 8x8 board?

for NxN board sizes larger than N=8, the known solutions are larger than N+1 of each color, lower bound O(N^2), i.e., a constant fraction of the board.  how many solutions are there for an NxN board?  (note well: N denotes the board size, not the number of queens.  in the traditional N queens problem, N could denote both, but in Peaceable Queens the numbers usually different.)

(for chess variants on larger boards, consider letting the sequence minus the number of initial queens define the initial number of pawns.)

the 8x8 solution of 9 queens each, diagrammed on A250000, can easily be changed to 8 queens and 1 king of each color, neither king in check.  but more elegant is to designate one of the 9 queens as royal: capture ends the game.  designate it with the queen piece; the other queens with pawns.  then play real chess starting from this "peaceable" start position.  bloody war likely breaks out very quickly.

while the sequence gives the number A = A250000(N) such that A white queens and A black queens fit on an NxN board, sometimes we can a fit a few more of one color, say, A+d white queens and A black queens.  one commenter on OEIS A250000 calls the situation "aggressors" and mentions that N = 15, A = 32, d = 2 is possible (without actually showing it).  we need another OEIS sequence giving the sequence of maximum d relative to A250000.

[dzsljhhg] short field

how would a sport change if distance from the one team's goal side to the other team's goal side were decreased?  basketball and soccer seem straightforward.  their endurance aspect will be decreased.

football requires adjusting kickoff location.  punting would be become rarer.

or long field.

[bjvanles] refrigerated truck

the common belief goes, you never know when you might need to murder, so it is good to own a gun, and so many people do (under the pretense "self defense", of course).  by the same logic, you never know when you might need to move a body, so it is good to own a vehicle with a body-sized freezer compartment.

such a vehicle allows body disposal to be widely separated in both time and location from that of the murder, making it difficult for investigators to connect a found body (if ever found) to an earlier disappearance.  it can also buy time for the perpetrator to carefully plan how to dispose of the body, especially useful if the murder was spur of the moment.

"real friends help you move a body."  if you have a real friend with such a vehicle, the person storing and eventually dumping the body can appear unrelated to the murderer, similar to contract killing (hitman).

an RV (camper van) seems a good form factor for such a vehicle.  it is a common type of vehicle so won't arouse suspicion.  (tangent: but maybe it will arouse suspicion if you are black, as camping seems not a popular activity among African-Americans.  of course, doing anything while black arouses suspicion in America.  this method of surreptitiously disposing a body is therefore only available to white folks.)  it likely already has an onboard generator to power appliances while the vehicle is parked.  very large, or extra, fuel tanks to support long-running power are likely an easily available option.  (though well-insulated freezers can be very energy efficient.)  driving an RV into remote wilderness to dump a body will also likely not arouse suspicion: such destinations are common sites for white-people camping.

with enough time, a body can become freeze dried, though I think this works well only in a freezer with automatic defrost, which loses efficiency.  after freeze drying, there won't be as much smelly decomposition when the body is returned to room temperature and less chance for bodily fluids to be deposited where forensic detectives might find them.

if you want to freeze dry (or desiccate by any other means), should you instead use a dedicated device?  unfortunately, many (most?) chest freezers do not have auto defrost.  avoid upright freezers in general, but especially avoid those with non-removeable shelves.  how easy is it to modify a freezer to freeze dry faster?  for food storage, people typically want to avoid freezer burn.

hypothesize that one of the reasons why the murdered corpse of Geraldine Kelly's husband stored for decades in an unpowered chest freezer at a storage facility did not arouse suspicion is because the freezer had originally been powered on for a long time, long enough to (somehow) freeze dry the body.

automatic defrost (presumably) deposits the evaporated (sublimated) vapors of freeze drying into the atmosphere outside the freezer.  the vapors are mostly water, but could incriminating trace chemicals deposit (desublimate or condense) on surfaces outside the freezer?  can forensic investigators distinguish between trace deposits from freeze drying legal meat versus long pig?  maybe it is better to avoid automatic defrost and keep everything sealed inside.

things left in a freezer for a long time get frozen to the bottom and sides.  this could be a problem if you suddenly need to dispose of the frozen body in a hurry.  maybe a layer of sand or some other powder can lessen sticking to the bottom.  or, wrap the body, e.g., tarp, drop cloth, window film, shower curtain, before freezing.  how large does the wrap need to be?  it needs to be ready in advance.

the remains of leaked and desublimated bodily fluids, including sand, other powder, or wrap, need to be cleaned out of the freezer, a time-consuming operation.  how difficult is it to thoroughly clean a freezer, for example, the inside of its drain hose if it has one?

given how useful RVs would be for "self defense", it is surprising that they are not as popular as guns (or maybe they are popular for this reason but I am unaware).  it is surprising they are not restricted and regulated like guns.  although restricting RVs and chest freezers will make a lot of non-murderers unhappy, making non-murderers unhappy has not stopped gun (and other weapon) regulators.

merely brandishing a gun is often assault, the crime of threatening to kill.  should conspicuously owning an RV be also?  can one inconspicuously own an RV?

how difficult is disposing of the body compared to all the other tasks required for getting away with murder?  study the murderers who did get caught.

also inspired by Benjamin Beale, New Orleans.

previously, burning instead of freezing.

[wtxbxaaq] Mendel vs virus

in an alternate universe, Gregor Mendel tries to understand genetics, but the traits he experiments on happen to be the result of viral infection, not what we in this universe call Mendelian genetics.  perhaps it is bad luck, or perhaps in the alternate universe, there are a lot more viruses that infect pea plants.  things go poorly.

[rjpcjamc] pawn on last rank

pawn allowed on 8th rank:

• N.  (orthodox chess: promotion forced)
• Y.
  1. pawn is allowed to move to 8th rank and remain a pawn:
     • N.  (the only way a pawn can be on 8th rank is that it must have started there, so this can only happen in compositions.  this is a chess variant in only the slightest sense.)
     • Y.
  2. pawn on 8th (could have started there, or moved there) is allowed to promote in a later turn:
     • N.  (moving a pawn to the 8th rank but voluntarily keeping it a completely useless pawn is therefore the ultimate underpromotion.  challenge: compose, probably stalemate.)
     • Y.  (delayed promotion)

5 possibilities.  note that 1 and 2 are orthogonal (Cartesian product).  2 of the possibilities (Y1N2N and Y1N2Y) are relevant only for compositions.  Y1Y2N is the composition challenge.

[xbsevhjo] men vs. wolves

some humans, annoyed at the threat of being eaten by wolves, decide to kill them, to exterminate one or more wolf packs in the area.  when throughout history and prehistory have these 2nd and 3rd best (land) predators on the planet (1st is ants) (the commonality among all three is hunting in coordinated packs) gone head to head?  nowadays, with one side able to use nuclear weapons and habitat destruction, the fight is too one-sided.  what is least amount of technology with which the humans have been victors?  what techniques did they employ?

(tangentially, it is easy to imagine Predator from the movies defeating a bear but struggling against a pack of wolves.  against (enough) ants, not a chance.)

inspired by Robinson Crusoe battling wolves in France.  though in that story, the wolves technically were not hunting the men, only their horses, if the Monday-morning wolf expert in the story is to be believed.

but hunting even the strongest predator is likely much easier than hunting the most elusive prey.  they know how to not to be killed, which is different from knowing how to kill.

previously, men versus dragons.

[npipkqow] violence: the cause of, and solution to, all the world's problems

what else?  in The Simpsons, "beer".

nuclear weapons, giant meteor, and other ways of extincting humanity would also solve all the world's problems, but it's hard to make the case that that any of them cause all the world's problems.

although violent revenge ("justice") for past violence is common, violence or a cycle of violence is usually a symptom of a deeper underlying problem, perhaps resource shortage.

sex.

[mdfjiozd] no bishop movement

chess variant removing ranged diagonal movement:

replace bishop with fairy ferz, or fairy ferz + fairy dabbaba.  the new piece is colorbound but not multiply colorbound (as unaugmented fairy dabbaba is).

replace queen with rook + fairy ferz.

but I think this makes the game worse because the interplay between ranged orthogonal movement and ranged diagonal movement makes orthodox chess interesting.

[kdvxzmmx] early all koroks

did anyone find all Koroks in Zelda BOTW before the first object map was published?

when was the total number first known?

vaguely related: if everyone had a different randomized universe, then there would be no universal map.

[bcmhtlmz] random convex polytope for linear programming

various tasks related to constructing a random D-dimensional linear programming problem:

we want a convex polyhedron (polytope) specified by a set of plane (hyperplane) inequalities (constraints).

for linear programming, sometimes it is OK for the polyhedron not to be closed, to have infinite volume.  but sometimes that will cause the solution to be infinite.  how can we detect an infinite polyhedron?  how can we determine if the solution is unbounded?

origin could be known to be an internal point (feasible).

are there any feasible points?

is a plane useless?  every point on the plane is infeasible.

all planes could be tangent to a sphere, or to an ellipsoid.

ellipsoid: start by constructing tangents to a sphere, then stretch and rotate.  probably need positive definite linear transformation.

hyperboloid and other quadrics could also work.  

plane equations could have all rational coefficients, so not all tangent to a sphere.  or could be: generalization of Pythagorean triples.

pick a random vertex.  might be tough in high dimensions.

given a subset of D plane equations (constraints), determine if their intersection is feasible.

start with a point at the intersection of D plane equations, then walk to an adjacent intersection, trying to move "inward" until reaching a feasible vertex.

pick a random vertex, project to sphere, and construct a tangent plane.  (previously.)

pick a random point on the sphere and construct tangent plane.  this is easy.

don't shave down all the way to tangency with the inscribed sphere, but a parallel plane only part of the way.  let the amount of material removed always be simple: just a pyramid.  is this always the case if maintaining convexity?  if starting from a random vertex, we probably need to know all its neighbor vertices.

pick a set of random points on a sphere and construct planes, but avoid tangent points from being too close to each other: Thomson problem.  does picking tangent points not close to each other tend to avoid vertices close to each other?  is there a better way to avoid vertices close to each other?

pick points from a spherical shell, construct planes perpendicular to the vector to the origin, then remove useless planes.  how thick should the shell be to avoid a lot of useless planes?  probably depends on number of dimensions.

rational points inside a spherical shell.  with rational planes close to being tangent.

if the polyhedron is nearly spherical, finding the maximum vertex might become easy.  if the feasible region is a sphere centered on the origin, the problem is likely very easy.  but in high dimensions, you probably don't have to worry about your polyhedron being too spherical: it probably takes an exponential number of planes to get nearly spherical.

convex hull.  worst case O(n^floor(d/2)) planes for n points, which is prohibitively huge in high dimensions.  should we, could we, avoid worst case?

limit to sparse plane equations: most coefficients are zero.

restrict to the positive (non-negative) quadrant (octant, etc.), i.e., all coordinates must be greater than or equal to zero.  restrict to a cube by requiring all coordinates to be less than or equal to some constant, or to a pyramid with x + y + z <= C.  more general pyramid: all positive coefficients.

we've ignored the objective function until now.  can we construct a random linear program whose solution is known in advance?

in quadratic programming, the maximum might not be at a vertex.  construct problems for which this is the case.  maximum could also be completely internal (probably makes the problem very easy), or on the boundary not at a vertex.

previous thoughts on generating random convex polytopes.

[ykierhyf] tilings of triangles and squares

tile the plane with squares and equilateral triangles all of the same side length.  this could be fun with physical tiles.

well investigated are the regular, semiregular, and uniform tilings, but random tilings also seem possible.

possible vertex configurations are (4 squares), (6 triangles), and (2 squares and 3 triangles).  2+3 has 2 possibilities: whether or not the squares are adjacent.

how careful do you need to be to avoid creating a gap into which nothing fits?

(as of early 2026, AI image generators failed to draw this random tiling.)

how difficult are such tilings to 4-color?  I think the following works: assign squares 2 colors.  2-color in a checkerboard pattern every connected component of just squares.  similarly, triangles with the other 2 colors.

this pair of tiles feels vaguely similar to Penrose tiles.  squares and triangles can tile periodically but they don't have to.

the edges form a penny graph.  these graphs feel aesthetically dense like the square and triangular grids.  the graph formed by the edges of the regular hexagon tiling does not feel aesthetically dense.

[kacgvdua] maximal Steiner tree

given a connected graph and a number N, select N vertices in the graph so that the size of the Steiner tree (Steiner tree problem on graphs) spanning the selected vertices is maximized.  (by definition, a Steiner tree minimizes its size, so this is a maximum-minimum problem.)  the size of a Steiner tree is sum of the weights of all its edges.  if unweighted edges, let every edge have unit weight.

if N=2, then the size of the maximum Steiner tree is graph diameter.  for larger N, this generalizes graph diameter: find the N points farthest away from each other.

for an adventure game exploring a graph, these are where the treasures are hidden.

what is the computational complexity of this problem?  solving the Steiner tree problem on graphs just once is NP-complete; now we are solving it binomial(number of graph vertices, N) times.

[ekzfaacc] first differences of prime indices of known Mersenne prime exponents

OEIS A135701

previously.

the lists below were computed from the known Mersenne primes as of March 2026.  if copying by hand, line numbers make it easier to rewrite a line correcting a mistake.  the last few entries might be incorrect because there might be more Mersenne primes in untested gaps before the largest known Mersenne prime.

in decimal:

1. 1
2. 1
3. 1
4. 2
5. 1
6. 1
7. 3
8. 7
9. 6
10. 4
11. 3
12. 67
13. 13
14. 96
15. 121
16. 11
17. 116
18. 128
19. 19
20. 594
21. 30
22. 131
23. 897
24. 181
25. 156
26. 2033
27. 3760
28. 2105
29. 1842
30. 6961
31. 41453
32. 7556
33. 28716
34. 9974
35. 108217
36. 3031
37. 256669
38. 402707
39. 452111
40. 179537
41. 113178
42. 258898
43. 126198
44. 263183
45. 313608
46. 26616
47. 833014
48. 906643
49. 166801
50. 294338
51. 2901904

in binary, substituting lower case letter o for zero to make the digits easier to distinguish by height (unless your font does lowercase numeral 1):

1. 1
1o. 1
11. 1
1oo. 1o
1o1. 1
11o. 1
111. 11
1ooo. 111
1oo1. 11o
1o1o. 1oo
1o11. 11
11oo. 1oooo11
11o1. 11o1
111o. 11ooooo
1111. 1111oo1
1oooo. 1o11
1ooo1. 111o1oo
1oo1o. 1ooooooo
1oo11. 1oo11
1o1oo. 1oo1o1oo1o
1o1o1. 1111o
1o11o. 1ooooo11
1o111. 111oooooo1
11ooo. 1o11o1o1
11oo1. 1oo111oo
11o1o. 1111111ooo1
11o11. 111o1o11oooo
111oo. 1ooooo111oo1
111o1. 111oo11oo1o
1111o. 11o11oo11ooo1
11111. 1o1oooo1111o11o1
1ooooo. 111o11oooo1oo
1oooo1. 111oooooo1o11oo
1ooo1o. 1oo11o1111o11o
1ooo11. 11o1oo11o1o111oo1
1oo1oo. 1o1111o1o111
1oo1o1. 11111o1o1o1oo111o1
1oo11o. 11ooo1oo1o1ooo1oo11
1oo111. 11o111oo11ooooo1111
1o1ooo. 1o1o1111o1o1o1ooo1
1o1oo1. 11o111o1oooo11o1o
1o1o1o. 111111oo11o1o1oo1o
1o1o11. 1111o11oo1111o11o
1o11oo. 1ooooooo1oooooo1111
1o11o1. 1oo11oo1oo1oooo1ooo
1o111o. 11oo11111111ooo
1o1111. 11oo1o11o1o11111o11o
11oooo. 11o111o1o1o11oo1oo11
11ooo1. 1o1ooo1o111oo1ooo1
11oo1o. 1ooo11111o111oooo1o
11oo11. 1o11ooo1ooo1111oo1oooo

791 bits including line numbers.  542 bits without.

[ksnmkfky] pixel glyphs

given a region partitioned into a small number of pixels not necessarily square (future post cmpvevta), enumerate glyphs.  two possible pixel colors, denoted background and foreground.  simplest enumeration: with N pixels, 2^N glyphs.

more complicated: avoid glyphs the same after translation.  avoid rotations, so text written sideways or upside down does not change meaning.  avoid reflections, making life easier for dyslexics.

only one connected component of foreground pixels.  strictly more (or less) than half the pixels are foreground.  or, avoid glyphs the same after inversion, swapping foreground and background.  (this might be useful for designing symbols for fairy chess, where we want both a black and white version of symbols.)

avoid not only "same" but also similar, within a given Hamming distance.  maximize the number of distinct symbols.

[gdyflnvq] movie 2: the sequel

brashly subtitle the sequel "the sequel".

[xocnwjgn] Antigone and Romeo and Juliet

it seems likely that Sophocles's Antigone, ending with two young lovers dead in a tomb, influenced Romeo and Juliet.  perhaps Shakespeare thought to himself, "I could write this better than Sophocles".

[yceqydqe] counting by scores

what if counting in English were base 20?  (inspired by French which is partially base 20.)

should 20-39 be "score" or "one score"?  can "and" be omitted?  is there a word for "four hundred"?

0 zero
1 one
2 two
3 three
4 four
5 five
6 six
7 seven
8 eight
9 nine
10 ten
11 eleven
12 twelve
13 thirteen
14 fourteen
15 fifteen
16 sixteen
17 seventeen
18 eighteen
19 nineteen
20 score
21 score and one
22 score and two
23 score and three
24 score and four
25 score and five
26 score and six
27 score and seven
28 score and eight
29 score and nine
30 score and ten
31 score and eleven
32 score and twelve
33 score and thirteen
34 score and fourteen
35 score and fifteen
36 score and sixteen
37 score and seventeen
38 score and eighteen
39 score and nineteen
40 two score
41 two score and one
42 two score and two
43 two score and three
44 two score and four
45 two score and five
46 two score and six
47 two score and seven
48 two score and eight
49 two score and nine
50 two score and ten
51 two score and eleven
52 two score and twelve
53 two score and thirteen
54 two score and fourteen
55 two score and fifteen
56 two score and sixteen
57 two score and seventeen
58 two score and eighteen
59 two score and nineteen
60 three score
61 three score and one
62 three score and two
63 three score and three
64 three score and four
65 three score and five
66 three score and six
67 three score and seven
68 three score and eight
69 three score and nine
70 three score and ten
71 three score and eleven
72 three score and twelve
73 three score and thirteen
74 three score and fourteen
75 three score and fifteen
76 three score and sixteen
77 three score and seventeen
78 three score and eighteen
79 three score and nineteen
80 four score
81 four score and one
82 four score and two
83 four score and three
84 four score and four
85 four score and five
86 four score and six
87 four score and seven (Lincoln's Gettysburg Address)
88 four score and eight
89 four score and nine
90 four score and ten
91 four score and eleven
92 four score and twelve
93 four score and thirteen
94 four score and fourteen
95 four score and fifteen
96 four score and sixteen
97 four score and seventeen
98 four score and eighteen
99 four score and nineteen
100 five score
101 five score and one
102 five score and two
103 five score and three
104 five score and four
105 five score and five
106 five score and six
107 five score and seven
108 five score and eight
109 five score and nine
110 five score and ten
111 five score and eleven
112 five score and twelve
113 five score and thirteen
114 five score and fourteen
115 five score and fifteen
116 five score and sixteen
117 five score and seventeen
118 five score and eighteen
119 five score and nineteen
120 six score
121 six score and one
122 six score and two
123 six score and three
124 six score and four
125 six score and five
126 six score and six
127 six score and seven
128 six score and eight
129 six score and nine
130 six score and ten
131 six score and eleven
132 six score and twelve
133 six score and thirteen
134 six score and fourteen
135 six score and fifteen
136 six score and sixteen
137 six score and seventeen
138 six score and eighteen
139 six score and nineteen
140 seven score
141 seven score and one
142 seven score and two
143 seven score and three
144 seven score and four
145 seven score and five
146 seven score and six
147 seven score and seven
148 seven score and eight
149 seven score and nine
150 seven score and ten
151 seven score and eleven
152 seven score and twelve
153 seven score and thirteen
154 seven score and fourteen
155 seven score and fifteen
156 seven score and sixteen
157 seven score and seventeen
158 seven score and eighteen
159 seven score and nineteen
160 eight score
161 eight score and one
162 eight score and two
163 eight score and three
164 eight score and four
165 eight score and five
166 eight score and six
167 eight score and seven
168 eight score and eight
169 eight score and nine
170 eight score and ten
171 eight score and eleven
172 eight score and twelve
173 eight score and thirteen
174 eight score and fourteen
175 eight score and fifteen
176 eight score and sixteen
177 eight score and seventeen
178 eight score and eighteen
179 eight score and nineteen
180 nine score
181 nine score and one
182 nine score and two
183 nine score and three
184 nine score and four
185 nine score and five
186 nine score and six
187 nine score and seven
188 nine score and eight
189 nine score and nine
190 nine score and ten
191 nine score and eleven
192 nine score and twelve
193 nine score and thirteen
194 nine score and fourteen
195 nine score and fifteen
196 nine score and sixteen
197 nine score and seventeen
198 nine score and eighteen
199 nine score and nineteen
200 ten score
201 ten score and one
202 ten score and two
203 ten score and three
204 ten score and four
205 ten score and five
206 ten score and six
207 ten score and seven
208 ten score and eight
209 ten score and nine
210 ten score and ten
211 ten score and eleven
212 ten score and twelve
213 ten score and thirteen
214 ten score and fourteen
215 ten score and fifteen
216 ten score and sixteen
217 ten score and seventeen
218 ten score and eighteen
219 ten score and nineteen
220 eleven score
221 eleven score and one
222 eleven score and two
223 eleven score and three
224 eleven score and four
225 eleven score and five
226 eleven score and six
227 eleven score and seven
228 eleven score and eight
229 eleven score and nine
230 eleven score and ten
231 eleven score and eleven
232 eleven score and twelve
233 eleven score and thirteen
234 eleven score and fourteen
235 eleven score and fifteen
236 eleven score and sixteen
237 eleven score and seventeen
238 eleven score and eighteen
239 eleven score and nineteen
240 twelve score
241 twelve score and one
242 twelve score and two
243 twelve score and three
244 twelve score and four
245 twelve score and five
246 twelve score and six
247 twelve score and seven
248 twelve score and eight
249 twelve score and nine
250 twelve score and ten
251 twelve score and eleven
252 twelve score and twelve
253 twelve score and thirteen
254 twelve score and fourteen
255 twelve score and fifteen
256 twelve score and sixteen
257 twelve score and seventeen
258 twelve score and eighteen
259 twelve score and nineteen
260 thirteen score
261 thirteen score and one
262 thirteen score and two
263 thirteen score and three
264 thirteen score and four
265 thirteen score and five
266 thirteen score and six
267 thirteen score and seven
268 thirteen score and eight
269 thirteen score and nine
270 thirteen score and ten
271 thirteen score and eleven
272 thirteen score and twelve
273 thirteen score and thirteen
274 thirteen score and fourteen
275 thirteen score and fifteen
276 thirteen score and sixteen
277 thirteen score and seventeen
278 thirteen score and eighteen
279 thirteen score and nineteen
280 fourteen score
281 fourteen score and one
282 fourteen score and two
283 fourteen score and three
284 fourteen score and four
285 fourteen score and five
286 fourteen score and six
287 fourteen score and seven
288 fourteen score and eight
289 fourteen score and nine
290 fourteen score and ten
291 fourteen score and eleven
292 fourteen score and twelve
293 fourteen score and thirteen
294 fourteen score and fourteen
295 fourteen score and fifteen
296 fourteen score and sixteen
297 fourteen score and seventeen
298 fourteen score and eighteen
299 fourteen score and nineteen
300 fifteen score
301 fifteen score and one
302 fifteen score and two
303 fifteen score and three
304 fifteen score and four
305 fifteen score and five
306 fifteen score and six
307 fifteen score and seven
308 fifteen score and eight
309 fifteen score and nine
310 fifteen score and ten
311 fifteen score and eleven
312 fifteen score and twelve
313 fifteen score and thirteen
314 fifteen score and fourteen
315 fifteen score and fifteen
316 fifteen score and sixteen
317 fifteen score and seventeen
318 fifteen score and eighteen
319 fifteen score and nineteen
320 sixteen score
321 sixteen score and one
322 sixteen score and two
323 sixteen score and three
324 sixteen score and four
325 sixteen score and five
326 sixteen score and six
327 sixteen score and seven
328 sixteen score and eight
329 sixteen score and nine
330 sixteen score and ten
331 sixteen score and eleven
332 sixteen score and twelve
333 sixteen score and thirteen
334 sixteen score and fourteen
335 sixteen score and fifteen
336 sixteen score and sixteen
337 sixteen score and seventeen
338 sixteen score and eighteen
339 sixteen score and nineteen
340 seventeen score
341 seventeen score and one
342 seventeen score and two
343 seventeen score and three
344 seventeen score and four
345 seventeen score and five
346 seventeen score and six
347 seventeen score and seven
348 seventeen score and eight
349 seventeen score and nine
350 seventeen score and ten
351 seventeen score and eleven
352 seventeen score and twelve
353 seventeen score and thirteen
354 seventeen score and fourteen
355 seventeen score and fifteen
356 seventeen score and sixteen
357 seventeen score and seventeen
358 seventeen score and eighteen
359 seventeen score and nineteen
360 eighteen score
361 eighteen score and one
362 eighteen score and two
363 eighteen score and three
364 eighteen score and four
365 eighteen score and five
366 eighteen score and six
367 eighteen score and seven
368 eighteen score and eight
369 eighteen score and nine
370 eighteen score and ten
371 eighteen score and eleven
372 eighteen score and twelve
373 eighteen score and thirteen
374 eighteen score and fourteen
375 eighteen score and fifteen
376 eighteen score and sixteen
377 eighteen score and seventeen
378 eighteen score and eighteen
379 eighteen score and nineteen
380 nineteen score
381 nineteen score and one
382 nineteen score and two
383 nineteen score and three
384 nineteen score and four
385 nineteen score and five
386 nineteen score and six
387 nineteen score and seven
388 nineteen score and eight
389 nineteen score and nine
390 nineteen score and ten
391 nineteen score and eleven
392 nineteen score and twelve
393 nineteen score and thirteen
394 nineteen score and fourteen
395 nineteen score and fifteen
396 nineteen score and sixteen
397 nineteen score and seventeen
398 nineteen score and eighteen
399 nineteen score and nineteen

previously, digits in base 20.

[wkoucwrm] French numbers base 20

numbers in French currently use a ugly mix of base 10 and base 20.  here, we consistently use base 20.  (we don't fix 11 through 19 to consistently use base 10 as done previously.)  future post: English in base 20.

1 un
2 deux
3 trois
4 quatre
5 cinq
6 six
7 sept
8 huit
9 neuf
10 dix
11 onze
12 douze
13 treize
14 quatorze
15 quinze
16 seize
17 dix-sept
18 dix-huit
19 dix-neuf
20 vingt
21 vingt-un
22 vingt-deux
23 vingt-trois
24 vingt-quatre
25 vingt-cinq
26 vingt-six
27 vingt-sept
28 vingt-huit
29 vingt-neuf
30 vingt-dix
31 vingt-onze
32 vingt-douze
33 vingt-treize
34 vingt-quatorze
35 vingt-quinze
36 vingt-seize
37 vingt-dix-sept
38 vingt-dix-huit
39 vingt-dix-neuf
40 deux-vingt
41 deux-vingt-un
42 deux-vingt-deux
43 deux-vingt-trois
44 deux-vingt-quatre
45 deux-vingt-cinq
46 deux-vingt-six
47 deux-vingt-sept
48 deux-vingt-huit
49 deux-vingt-neuf
50 deux-vingt-dix
51 deux-vingt-onze
52 deux-vingt-douze
53 deux-vingt-treize
54 deux-vingt-quatorze
55 deux-vingt-quinze
56 deux-vingt-seize
57 deux-vingt-dix-sept
58 deux-vingt-dix-huit
59 deux-vingt-dix-neuf
60 trois-vingt
61 trois-vingt-un
62 trois-vingt-deux
63 trois-vingt-trois
64 trois-vingt-quatre
65 trois-vingt-cinq
66 trois-vingt-six
67 trois-vingt-sept
68 trois-vingt-huit
69 trois-vingt-neuf
70 trois-vingt-dix
71 trois-vingt-onze
72 trois-vingt-douze
73 trois-vingt-treize
74 trois-vingt-quatorze
75 trois-vingt-quinze
76 trois-vingt-seize
77 trois-vingt-dix-sept
78 trois-vingt-dix-huit
79 trois-vingt-dix-neuf
80 quatre-vingt
81 quatre-vingt-un
82 quatre-vingt-deux
83 quatre-vingt-trois
84 quatre-vingt-quatre
85 quatre-vingt-cinq
86 quatre-vingt-six
87 quatre-vingt-sept
88 quatre-vingt-huit
89 quatre-vingt-neuf
90 quatre-vingt-dix
91 quatre-vingt-onze
92 quatre-vingt-douze
93 quatre-vingt-treize
94 quatre-vingt-quatorze
95 quatre-vingt-quinze
96 quatre-vingt-seize
97 quatre-vingt-dix-sept
98 quatre-vingt-dix-huit
99 quatre-vingt-dix-neuf
100 cinq-vingt
101 cinq-vingt-un
102 cinq-vingt-deux
103 cinq-vingt-trois
104 cinq-vingt-quatre
105 cinq-vingt-cinq
106 cinq-vingt-six
107 cinq-vingt-sept
108 cinq-vingt-huit
109 cinq-vingt-neuf
110 cinq-vingt-dix
111 cinq-vingt-onze
112 cinq-vingt-douze
113 cinq-vingt-treize
114 cinq-vingt-quatorze
115 cinq-vingt-quinze
116 cinq-vingt-seize
117 cinq-vingt-dix-sept
118 cinq-vingt-dix-huit
119 cinq-vingt-dix-neuf
120 six-vingt
121 six-vingt-un
122 six-vingt-deux
123 six-vingt-trois
124 six-vingt-quatre
125 six-vingt-cinq
126 six-vingt-six
127 six-vingt-sept
128 six-vingt-huit
129 six-vingt-neuf
130 six-vingt-dix
131 six-vingt-onze
132 six-vingt-douze
133 six-vingt-treize
134 six-vingt-quatorze
135 six-vingt-quinze
136 six-vingt-seize
137 six-vingt-dix-sept
138 six-vingt-dix-huit
139 six-vingt-dix-neuf
140 sept-vingt
141 sept-vingt-un
142 sept-vingt-deux
143 sept-vingt-trois
144 sept-vingt-quatre
145 sept-vingt-cinq
146 sept-vingt-six
147 sept-vingt-sept
148 sept-vingt-huit
149 sept-vingt-neuf
150 sept-vingt-dix
151 sept-vingt-onze
152 sept-vingt-douze
153 sept-vingt-treize
154 sept-vingt-quatorze
155 sept-vingt-quinze
156 sept-vingt-seize
157 sept-vingt-dix-sept
158 sept-vingt-dix-huit
159 sept-vingt-dix-neuf
160 huit-vingt
161 huit-vingt-un
162 huit-vingt-deux
163 huit-vingt-trois
164 huit-vingt-quatre
165 huit-vingt-cinq
166 huit-vingt-six
167 huit-vingt-sept
168 huit-vingt-huit
169 huit-vingt-neuf
170 huit-vingt-dix
171 huit-vingt-onze
172 huit-vingt-douze
173 huit-vingt-treize
174 huit-vingt-quatorze
175 huit-vingt-quinze
176 huit-vingt-seize
177 huit-vingt-dix-sept
178 huit-vingt-dix-huit
179 huit-vingt-dix-neuf
180 neuf-vingt
181 neuf-vingt-un
182 neuf-vingt-deux
183 neuf-vingt-trois
184 neuf-vingt-quatre
185 neuf-vingt-cinq
186 neuf-vingt-six
187 neuf-vingt-sept
188 neuf-vingt-huit
189 neuf-vingt-neuf
190 neuf-vingt-dix
191 neuf-vingt-onze
192 neuf-vingt-douze
193 neuf-vingt-treize
194 neuf-vingt-quatorze
195 neuf-vingt-quinze
196 neuf-vingt-seize
197 neuf-vingt-dix-sept
198 neuf-vingt-dix-huit
199 neuf-vingt-dix-neuf
200 dix-vingt
201 dix-vingt-un
202 dix-vingt-deux
203 dix-vingt-trois
204 dix-vingt-quatre
205 dix-vingt-cinq
206 dix-vingt-six
207 dix-vingt-sept
208 dix-vingt-huit
209 dix-vingt-neuf
210 dix-vingt-dix
211 dix-vingt-onze
212 dix-vingt-douze
213 dix-vingt-treize
214 dix-vingt-quatorze
215 dix-vingt-quinze
216 dix-vingt-seize
217 dix-vingt-dix-sept
218 dix-vingt-dix-huit
219 dix-vingt-dix-neuf
220 onze-vingt
221 onze-vingt-un
222 onze-vingt-deux
223 onze-vingt-trois
224 onze-vingt-quatre
225 onze-vingt-cinq
226 onze-vingt-six
227 onze-vingt-sept
228 onze-vingt-huit
229 onze-vingt-neuf
230 onze-vingt-dix
231 onze-vingt-onze
232 onze-vingt-douze
233 onze-vingt-treize
234 onze-vingt-quatorze
235 onze-vingt-quinze
236 onze-vingt-seize
237 onze-vingt-dix-sept
238 onze-vingt-dix-huit
239 onze-vingt-dix-neuf
240 douze-vingt
241 douze-vingt-un
242 douze-vingt-deux
243 douze-vingt-trois
244 douze-vingt-quatre
245 douze-vingt-cinq
246 douze-vingt-six
247 douze-vingt-sept
248 douze-vingt-huit
249 douze-vingt-neuf
250 douze-vingt-dix
251 douze-vingt-onze
252 douze-vingt-douze
253 douze-vingt-treize
254 douze-vingt-quatorze
255 douze-vingt-quinze
256 douze-vingt-seize
257 douze-vingt-dix-sept
258 douze-vingt-dix-huit
259 douze-vingt-dix-neuf
260 treize-vingt
261 treize-vingt-un
262 treize-vingt-deux
263 treize-vingt-trois
264 treize-vingt-quatre
265 treize-vingt-cinq
266 treize-vingt-six
267 treize-vingt-sept
268 treize-vingt-huit
269 treize-vingt-neuf
270 treize-vingt-dix
271 treize-vingt-onze
272 treize-vingt-douze
273 treize-vingt-treize
274 treize-vingt-quatorze
275 treize-vingt-quinze
276 treize-vingt-seize
277 treize-vingt-dix-sept
278 treize-vingt-dix-huit
279 treize-vingt-dix-neuf
280 quatorze-vingt
281 quatorze-vingt-un
282 quatorze-vingt-deux
283 quatorze-vingt-trois
284 quatorze-vingt-quatre
285 quatorze-vingt-cinq
286 quatorze-vingt-six
287 quatorze-vingt-sept
288 quatorze-vingt-huit
289 quatorze-vingt-neuf
290 quatorze-vingt-dix
291 quatorze-vingt-onze
292 quatorze-vingt-douze
293 quatorze-vingt-treize
294 quatorze-vingt-quatorze
295 quatorze-vingt-quinze
296 quatorze-vingt-seize
297 quatorze-vingt-dix-sept
298 quatorze-vingt-dix-huit
299 quatorze-vingt-dix-neuf
300 quinze-vingt
301 quinze-vingt-un
302 quinze-vingt-deux
303 quinze-vingt-trois
304 quinze-vingt-quatre
305 quinze-vingt-cinq
306 quinze-vingt-six
307 quinze-vingt-sept
308 quinze-vingt-huit
309 quinze-vingt-neuf
310 quinze-vingt-dix
311 quinze-vingt-onze
312 quinze-vingt-douze
313 quinze-vingt-treize
314 quinze-vingt-quatorze
315 quinze-vingt-quinze
316 quinze-vingt-seize
317 quinze-vingt-dix-sept
318 quinze-vingt-dix-huit
319 quinze-vingt-dix-neuf
320 seize-vingt
321 seize-vingt-un
322 seize-vingt-deux
323 seize-vingt-trois
324 seize-vingt-quatre
325 seize-vingt-cinq
326 seize-vingt-six
327 seize-vingt-sept
328 seize-vingt-huit
329 seize-vingt-neuf
330 seize-vingt-dix
331 seize-vingt-onze
332 seize-vingt-douze
333 seize-vingt-treize
334 seize-vingt-quatorze
335 seize-vingt-quinze
336 seize-vingt-seize
337 seize-vingt-dix-sept
338 seize-vingt-dix-huit
339 seize-vingt-dix-neuf
340 dix-sept-vingt
341 dix-sept-vingt-un
342 dix-sept-vingt-deux
343 dix-sept-vingt-trois
344 dix-sept-vingt-quatre
345 dix-sept-vingt-cinq
346 dix-sept-vingt-six
347 dix-sept-vingt-sept
348 dix-sept-vingt-huit
349 dix-sept-vingt-neuf
350 dix-sept-vingt-dix
351 dix-sept-vingt-onze
352 dix-sept-vingt-douze
353 dix-sept-vingt-treize
354 dix-sept-vingt-quatorze
355 dix-sept-vingt-quinze
356 dix-sept-vingt-seize
357 dix-sept-vingt-dix-sept
358 dix-sept-vingt-dix-huit
359 dix-sept-vingt-dix-neuf
360 dix-huit-vingt
361 dix-huit-vingt-un
362 dix-huit-vingt-deux
363 dix-huit-vingt-trois
364 dix-huit-vingt-quatre
365 dix-huit-vingt-cinq
366 dix-huit-vingt-six
367 dix-huit-vingt-sept
368 dix-huit-vingt-huit
369 dix-huit-vingt-neuf
370 dix-huit-vingt-dix
371 dix-huit-vingt-onze
372 dix-huit-vingt-douze
373 dix-huit-vingt-treize
374 dix-huit-vingt-quatorze
375 dix-huit-vingt-quinze
376 dix-huit-vingt-seize
377 dix-huit-vingt-dix-sept
378 dix-huit-vingt-dix-huit
379 dix-huit-vingt-dix-neuf
380 dix-neuf-vingt
381 dix-neuf-vingt-un
382 dix-neuf-vingt-deux
383 dix-neuf-vingt-trois
384 dix-neuf-vingt-quatre
385 dix-neuf-vingt-cinq
386 dix-neuf-vingt-six
387 dix-neuf-vingt-sept
388 dix-neuf-vingt-huit
389 dix-neuf-vingt-neuf
390 dix-neuf-vingt-dix
391 dix-neuf-vingt-onze
392 dix-neuf-vingt-douze
393 dix-neuf-vingt-treize
394 dix-neuf-vingt-quatorze
395 dix-neuf-vingt-quinze
396 dix-neuf-vingt-seize
397 dix-neuf-vingt-dix-sept
398 dix-neuf-vingt-dix-huit
399 dix-neuf-vingt-dix-neuf

[agzcybrb] counting to 100 in French the right way

1 un
2 deux
3 trois
4 quatre
5 cinq
6 six
7 sept
8 huit
9 neuf
10 dix
11 dix un
12 dix deux
13 dix trois
14 dix quatre
15 dix cinq
16 dix six
17 dix sept
18 dix huit
19 dix neuf
20 deux dix
21 deux dix un
22 deux dix deux
23 deux dix trois
24 deux dix quatre
25 deux dix cinq
26 deux dix six
27 deux dix sept
28 deux dix huit
29 deux dix neuf
30 trois dix
31 trois dix un
32 trois dix deux
33 trois dix trois
34 trois dix quatre
35 trois dix cinq
36 trois dix six
37 trois dix sept
38 trois dix huit
39 trois dix neuf
40 quatre dix
41 quatre dix un
42 quatre dix deux
43 quatre dix trois
44 quatre dix quatre
45 quatre dix cinq
46 quatre dix six
47 quatre dix sept
48 quatre dix huit
49 quatre dix neuf
50 cinq dix
51 cinq dix un
52 cinq dix deux
53 cinq dix trois
54 cinq dix quatre
55 cinq dix cinq
56 cinq dix six
57 cinq dix sept
58 cinq dix huit
59 cinq dix neuf
60 six dix
61 six dix un
62 six dix deux
63 six dix trois
64 six dix quatre
65 six dix cinq
66 six dix six
67 six dix sept
68 six dix huit
69 six dix neuf
70 sept dix
71 sept dix un
72 sept dix deux
73 sept dix trois
74 sept dix quatre
75 sept dix cinq
76 sept dix six
77 sept dix sept
78 sept dix huit
79 sept dix neuf
80 huit dix
81 huit dix un
82 huit dix deux
83 huit dix trois
84 huit dix quatre
85 huit dix cinq
86 huit dix six
87 huit dix sept
88 huit dix huit
89 huit dix neuf
90 neuf dix
91 neuf dix un
92 neuf dix deux
93 neuf dix trois
94 neuf dix quatre
95 neuf dix cinq
96 neuf dix six
97 neuf dix sept
98 neuf dix huit
99 neuf dix neuf
100 cent

this isn't too unrealistic: all we need is for the next entity that conquers France to fix their language for them.  it seems like the kind of thing the Nazis would have done, imposing order, though maybe they would have just imposed German.  language reform by brutality has precedent in France, most notably previously done by Caesar who came saw conquered then imposed Latin, which stuck and evolved into French, and more recently by vergonha.  maybe France surrenders this time because they flub their numeric nuclear launch codes.

when this happens, French would join Japanese in having a perfectly regular counting up to 99.  (for others, see https://web.archive.org/web/20200303062737/http://www.sf.airnet.ne.jp/ts/language/number.html) and because French already does long scale 1000000^Latin for large numbers (more regular than short scale 1000^(Latin+1) as done in American), this would catapult them from having one of the worst words for numbers to one of the best.  (incidentally, both scales have already fixed the worst part of Latin numbers by preferring octodecillion and novemdecillion/novendecillion over duodevigintillion and undevigintillion.)

previously, fixing English similarly.

[foyujkfq] digits of base 20

we need 10 additional characters for digits of base 20.  let's choose them from the letters as hexadecimal does, but avoid similar looking characters:

A a similar to 4
B b similar to 8 and 6
C c 10
D d similar to 0
E e 11
F f 12
G g similar to 6 and 9
H h 13
I i similar to 1
J j 14
K k 15
L l similar to 1
M m 16
N n similar to h
O o similar to 0
P p similar to h
Q q similar to 0 and 9
R r similar to h
S s similar to 5
T t similar to J
U u 17
V v similar to U and R
W w 18
X x 19
Y y similar to X
Z z similar to 2

so

0 1 2 3 4 5 6 7 8 9
C E F H J K M U W X
c e f h j k m u w x

dyslexics might dislike having both 3 and E, but we are not fixing 2 and 5, nor the non-dyslexic similar characters 1 and 7, and 4 and 9.

lowercase h p n r v are all handwritten with similar motion, so we only include h.

other posts: base 20 English and French.  Plus Codes.

[knlwtvtb] fixing English numbers

we make the words for English numbers regular up to 99.  (future post French.)

0 zero
1 one
2 two
3 three
4 four
5 five
6 six
7 seven
8 eight
9 nine
10 one-ty-zero
11 one-ty-one
12 one-ty-two
13 one-ty-three
14 one-ty-four
15 one-ty-five
16 one-ty-six
17 one-ty-seven
18 one-ty-eight
19 one-ty-nine
20 two-ty-zero
21 two-ty-one
22 two-ty-two
23 two-ty-three
24 two-ty-four
25 two-ty-five
26 two-ty-six
27 two-ty-seven
28 two-ty-eight
29 two-ty-nine
30 three-ty-zero
31 three-ty-one
32 three-ty-two
33 three-ty-three
34 three-ty-four
35 three-ty-five
36 three-ty-six
37 three-ty-seven
38 three-ty-eight
39 three-ty-nine
40 four-ty-zero
41 four-ty-one
42 four-ty-two
43 four-ty-three
44 four-ty-four
45 four-ty-five
46 four-ty-six
47 four-ty-seven
48 four-ty-eight
49 four-ty-nine
50 five-ty-zero
51 five-ty-one
52 five-ty-two
53 five-ty-three
54 five-ty-four
55 five-ty-five
56 five-ty-six
57 five-ty-seven
58 five-ty-eight
59 five-ty-nine
60 six-ty-zero
61 six-ty-one
62 six-ty-two
63 six-ty-three
64 six-ty-four
65 six-ty-five
66 six-ty-six
67 six-ty-seven
68 six-ty-eight
69 six-ty-nine
70 seven-ty-zero
71 seven-ty-one
72 seven-ty-two
73 seven-ty-three
74 seven-ty-four
75 seven-ty-five
76 seven-ty-six
77 seven-ty-seven
78 seven-ty-eight
79 seven-ty-nine
80 eight-ty-zero
81 eight-ty-one
82 eight-ty-two
83 eight-ty-three
84 eight-ty-four
85 eight-ty-five
86 eight-ty-six
87 eight-ty-seven
88 eight-ty-eight
89 eight-ty-nine
90 nine-ty-zero
91 nine-ty-one
92 nine-ty-two
93 nine-ty-three
94 nine-ty-four
95 nine-ty-five
96 nine-ty-six
97 nine-ty-seven
98 nine-ty-eight
99 nine-ty-nine

optionally omit "-zero" in multiples of 10 for brevity, if it does not cause ambiguity.  for example, if reading a string of numbers, we need to keep "-zero" to distinguish "90 9" from "99", or "10 4" from "14".  (the former is a problem in unfixed English, additionally ambiguous with "19 9".)

keeping "ten" as an isolated irregular but convenient synonym for one-ty-zero seems reasonable.

one-ty sounds similar to the old word twenty, but not the new word two-ty.

[argeufrf] automatic field separator

unless this is to be first field (e.g., word) in the line (record), print the field separator (e.g., space for words in a line, comma for CSV, tab for TSV).  then, print the given field data.  the output subsystem automatically keeps track of whether we are at first field in a record.

optionally, automatic escaping or quoting if the field data contains a field or record separator.

similarly, automatic record terminator (e.g., newline) on orderly file close.

in general: field prefix, field terminator, field separator, record prefix, record terminator, record separator.  not all of these will be used.

[lzlpplvi] shortest path on a Euclidean manifold with wormholes

in Euclidean space, the shortest path between two points is a line segment.  then, modify the space:

designate two special points for which travel between them is free, costs no distance, i.e., a wormhole connects the points.  then, the shortest path between two points might no longer be the line between them; it might be to travel to and from the wormhole mouths.

(which pairs of points are affected?  this is a problem in 2*D dimensional space.)

instead of just two special points, add a line of special points with free travel between any two points on the line.  shortest paths will often involve traveling orthogonally to the line, traveling along the line for free, then departing orthogonally from the line to the destination.  maybe this is a river in which transportation cost is negligible.  or, minimize walking where there are moving sidewalks.

the wormhole mouths and river line are examples of sets of points.  a zone of free travel can be generalized to any set of points.  one could have multiple zones: free travel within each zone, but, unless zones are adjacent, you have to exit a zone and travel through the overworld to connect to other zones.

how difficult is finding the shortest path?

once we have a notion of shortest path (geodesic), we can define minimum spanning tree, traveling salesman, Euclidean Steiner tree, Voronoi diagram, Delauney triangulation.

[jxrlppkn] most defenses

consider a mate in N chess problem.  how many paths through the game tree end in checkmate in exactly N moves?  (also consider less than N moves.)

for a good chess problem with no duals, every attacker move in that subset of the game tree has exactly one move that accomplishes mate in N.  if not, specify a strategy (e.g., fewest defender branches) that deterministically picks one attacker move and prune the rest of the attacker branches.  only alternatives by the defender contribute to the total branches.

compose chess problems that maximize the number of paths.  or the number of nodes in the pruned game graph (branches may merge through transposition).

such problems tend to be annoying for humans to solve, needing to check many cases.

[dlaexadi] abstract swordfighting

incomplete information version:

at the beginning of a turn, you see your opponent's attack.  you choose a defense, which is revealed to the opponent.  attack and defense jointly define the probability that the opponent's attack is successful, and the distribution is sampled.  if the attack succeeds, you lose the game.  if the attack fails, it is your turn to choose an attack.  the defense you chose earlier constrains your possibilities of attack.  the attack you choose will constrain your possibilities of defense next turn if the game continues.

imperfect information version:

at the beginning of a turn, the opponent chooses attack part 1 and attack part 2.  you see part 1 of your opponent's attack.  you choose a defense, then part 2 of the opponent's attack is revealed.  your defense is revealed to the opponent.  your defense and opponent's attack part 2 deterministically define whether the opponent's attack succeeds.  if the attack succeeds, you lose the game.  if the attack fails, you choose your attacks part 1 and part 2, and the opponent chooses a defense after seeing your part 1.  your defense earlier constrains your attack part 1, attack part 1 constrains attack part 2, and attack part 2 constrains defense next move if the game continues.

perfect and complete information version:

there is only one possible part 2 attack following any given part 1 attack (so there is no reason to conceal part 2).  the way to win the game is to maneuver the opponent into a position such that there is no defense against an attack.

simplified imperfect information version:

there is only one possible part 1 attack, so it provides no information.  the information available to the defender is the attacker's defense last move.

fancier imperfect information version:

both attack and defense have a "to be revealed" portion and a portion not revealed that also constrains actions next move.  "to be revealed" contains enough information to score the current move, to determine if there is a winner.  we need a mechanism to force a player to obey self-imposed constraints hidden from the other player.  complicated sequences over many moves can be encoded as strategies as done in game theory.  such sequences could involve changing internal state, not affecting the outcome of the current move.

there are a whole bunch of tables that need to be defined: what attacks can follow what defenses and vice versa, and the probabilistic or deterministic outcomes of attack versus defense.  defining these tables offers infinite possibilities of game design.  for example, an action taken one move can constrain actions many moves down the line.

I think all two-player non-cooperative games can be expressed in this framework.

[rwlckvhw] Make Genocide Great Again

"I support Netanyahu, W, and Hitler."

(three flags melded into one)

"For the safety and security of our people."

(in three languages)

(should it be Andrew Jackson instead of George W. Bush?  (¿por qué no los dos?  porque the comedy rule of three.)  was "safety and security for us white colonizers" touted at the time as the justification for genocide of Native Americans?  probably yes, left over as the "cowboys vs Indians" children's game, among other things.  going further into the past also finds many more great</sarcasm> genociders: Khan(s), Caesar(s),...  where is the chronological boundary for "Again"?)

"Made Genocide Great Again" in light blue on white, framing a smiling portrait of Netanyahu.  or, wearing a MGGA hat.

"we are not like Hitler.  we are good people, doing good genocide humanitarian crisis."

[lyncpmou] expected number of barriers passed

let barrier(n,h) be the probability of passing barrier number n, starting from n=1.  h is a parameter that will be used later.

to pass exactly n barriers, one needs to consecutively pass barriers 1 through n, then fail barrier n+1:

exactly(n,h)= prod(i=1,n, barrier(i,h)) * (1-barrier(n+1,h))

(Pari/GP syntax)

the expected number of barriers crossed before the first failure is an infinite sum.  here is the partial sum:

partialexpected(n,h)= sum(i=1,n, i*exactly(i,h))

first, let us investigate the barrier probability being a constant, not dependent on n:

barrier(n,h)= h

the expected number E of barriers crossed for constant barrier probability h:

h=1/6 E=1/5
h=1/5 E=1/4
h=1/4 E=1/3
h=1/3 E=1/2
h=2/5 E=2/3
h=1/2 E=1
h=3/5 E=3/2
h=2/3 E=2
h=3/4 E=3
h=4/5 E=4
h=5/6 E=5

the above exact values are conjectures from the partial sum, e.g.:

partialexpected(999,5.0/6)
4.9999999999999999999999999999999999940
cpu time = 107 ms, real time = 107 ms.

a closed-form formula for a given constant barrier probability is probably not difficult to derive, probably something like geometric series.

if the barrier probability is 1/n, the expected number of barriers crossed is e-1:

barrier(n,h)= 1.0/n
partialexpected(99)
1.7182818284590452353602874713526624977
partialexpected(98)
1.7182818284590452353602874713526624977

the second calculation of only 98 terms verifies convergence of the partial sum.  a barrier probability decreasing as a function of n is nice for making the partial sum converge quickly.

(in Pari/GP, if a function argument is omitted -- in this case h -- it is assigned a default value of 0.)

if the barrier probability is 1/(2*n), equivalently 0.5/n, then the expected value is exp(0.5)-1 (via the RIES Inverse Symbolic Calculator). (the original Inverse Symbolic Calculator has gone offline.)

barrier(n,h)= h/n
partialexpected(99,0.5)
0.64872127070012814684865078781416357161

we conjecture that the expected value for this form of barrier probability is exp(h)-1.  therefore, if h = log(2) ~= 0.693, the expected value is 1 :

partialexpected(99,log(2))
0.99999999999999999999999999999999999996

let the barrier probability decay exponentially:

barrier(n,h)= h^n

if the barrier probability is 1/2^n , then the expected value has the following value, unknown if it has a closed form:

partialexpected(99,0.5)
0.64163256065515386629384277022542943422

if h = 0.7 , the expected value is greater than 1 :

partialexpected(99,0.7)
1.1942513927487148419874371583834978589

we use Pari/GP's nifty solve function to find the value of h such that the expected value is exactly 1 .  does this value of h have a closed form expression?

h = solve(x=0.5,0.7, partialexpected(99,x)-1)
0.64522270323602097913425166394402633225
partialexpected(99,h)
0.99999999999999999999999999999999999996

recall that if the barrier probability is 1/n, the expected value is e-1 :

barrier(n,h)= 1.0/(n+h)
partialexpected(99,0)
1.7182818284590452353602874713526624977

the first barrier is passed with probability 1, not a barrier at all, a free +1 to the expected value.  if we shift all barriers by one, then the expected value is unsurprisingly e-2 :

partialexpected(99,1)
0.71828182845904523536028747135266249771

this is the amount of shift that makes the expected value 1 :

h= solve(x=0,1, partialexpected(99,x)-1)
0.53571808926964201274443895890553620494

let the barrier probability obey a power law.  we shift by 1 to skip the free first barrier:

barrier(n,h)= (n+1.0)^h

when the exponent is -1, the expected value is e-2 as noted before:

partialexpected(99,-1)
0.71828182845904523536028747135266249771

if the probability is 1/sqrt(n+1) , the expected value is greater than 1 :

partialexpected(99,-0.5)
1.4695063145210475624756367446601502576

this exponent makes the expected value 1 :

h= solve(x=-0.5,-1, partialexpected(99,x)-1)
-0.73888040289770820386751815624320497895

inspired by modeling family sizes.  have children until you can't, perhaps because of death caused by complications from childbirth.

if sampling a random tree where the number of children of each node has expected value 1, the population stays approximately constant generation to generation (I think).  such a tree models asexual reproduction, or a filtered view of sexual reproduction paying attention to only one gender.

[jnqlnxts] classification of extensive-form games by computational requirements

game length:

1. infinite-length games possible.  (if finite game state, then game must be loopy.)
2. games are guaranteed to be finite length but finite games of arbitrary length possible.  (e.g., players take turns naming smaller positive integers, Kirby-Paris hydra.)
3. maximum game length known at start.

this classification is practically fuzzy: maximum game length could be finite but so large as to effectively be infinite (future post hxfdvhup).  (e.g., loopy games with special rules prohibiting repetition.)  or, infinite or extremely long games are theoretically possible but require both players to be silly.

game state:

bounded or unbounded amount of information needed to describe game state.

common case of unbounded: loopy game has special rules regarding repeating a previous position (e.g., chess 3-fold repetition, go 囲碁 super-ko) so, as the game progresses, game state includes a growing list of positions already seen (in practice, a hash table).

this classification is practically fuzzy: theoretically bounded but so huge as to effectively be infinite.  chess and go 囲碁 have finite numbers of possible positions so keeping track of all of them classifying each as already seen or not requires only a finite bounded (but huge) amount of information.

the above classifications matter for computer implementations of games: how many bits wide should the move counter be to avoid overflow?

other famous classifications: perfect vs. imperfect information (players hide information).  complete vs. incomplete information (randomness or chance).

[tgtdtucz] Fourier series for ellipse in polar coordinates

below is the formula for an ellipse in polar coordinates, with eccentricity e and one focus at the origin (multiply by scaling factor to embiggen):

r = 1 / (1 - e*cos(t))

the formula has cosine in denominator.  we compute its Fourier transform to get a formula with the sines and cosines only in the numerators.  (only cos is needed.)

Mathematica:

FourierCosSeries[1/(1-e*Cos[t]), t, 7, Assumptions-> e < 1 && 0 <= e] //Timing

{245.35853, 1/Sqrt[1 - e^2] + (2*(-1 + 1/Sqrt[1 - e^2])*Cos[t])/e - (2*(-2 + e^2 + 2*Sqrt[1 - e^2])*Cos[2*t])/(e^2*Sqrt[1 - e^2]) + (2*(-4 + 4/Sqrt[1 - e^2] + e^2*(1 - 3/Sqrt[1 - e^2]))*Cos[3*t])/e^3 + (2*(8 + e^4 - 8*Sqrt[1 - e^2] + 4*e^2*(-2 + Sqrt[1 - e^2]))*Cos[4*t])/(e^4*Sqrt[1 - e^2]) - (2*(16 - 16/Sqrt[1 - e^2] + e^4*(1 - 5/Sqrt[1 - e^2]) + e^2*(-12 + 20/Sqrt[1 - e^2]))*Cos[5*t])/e^5 - (2*(e^6 + e^2*(48 - 32*Sqrt[1 - e^2]) + 6*e^4*(-3 + Sqrt[1 - e^2]) + 32*(-1 + Sqrt[1 - e^2]))*Cos[6*t])/(e^6*Sqrt[1 - e^2]) + (2*(e^6*(-7 + Sqrt[1 - e^2]) - 64*(-1 + Sqrt[1 - e^2]) - 8*e^4*(-7 + 3*Sqrt[1 - e^2]) + 16*e^2*(-7 + 5*Sqrt[1 - e^2]))*Cos[7*t])/(e^7*Sqrt[1 - e^2])}

what is a general formula for the coefficients (to get an infinite series)?  this has almost certainly already been solved, but I could not find such a formula published anywhere.  it seems like the kind of thing Joseph Fourier might have done himself.

investigate Fourier transforms of generalizations of the original formula with cosine in the denominator, for example, r(theta) = 1/(sum(n, A[n] * cos(n*theta + B[n]))).  what is the Fourier transform of the reciprocal of a Fourier transform?  previously, polar plots of regular (not reciprocal) Fourier series.  under what conditions is the shape convex?

[hgulalrr] subtracting times of day with units

time elapsed between:
4th day of the month 11:59:59 PM
6th day of the month 6:06:06 AM

$ units
You have: (6 day + 6 hr + 6 min + 6 s) - (4 day + 23 hr + 59 min + 59 s)
You want: day;hr;min;s
1 day + 6 hr + 6 min + 7 s

the semicolon syntax is documented in the units man page in the section entitled "UNIT LISTS: CONVERSION TO SUMS OF UNITS".

computing elapsed time requires more effort if endpoints are in different months, e.g., January 31, February 1 = January 32, February n = January (31 + n),... March 1 = February ( {28 or 29} + 1) = January (31 + {28 or 29} + 1).

incorrect if Daylight Saving Time changes within the interval; incorrect if endpoints are in different time zones; incorrect if a leap second occurs; incorrect if crossing the boundary between Gregorian and Julian calendars.

for such complicated cases, first convert each endpoint into a format which increases continuously and uniformly, then subtract, then convert with units to "day;hour;minute;second".

note well that Unix time does NOT increase uniformly: it stops at a leap second (or more precisely, it discontinuously decreases by one second).  subtracting two Unix times yields an elapsed time value that is incorrect by the number of leap seconds which occurred during the interval.

use the "right" counter instead of POSIX Unix time.  note that, as of Debian Trixie, you need to install the tzdata-legacy package to get the "right" counter value of a given date and time.

[gjjimaii] estimating pi with primes

we compute an approximation of Riemann zeta(2) = pi^2/6 using the Euler product formula over primes, then solve for pi:

? for(n=0, 10, z=1; forprime(p=1, 10^n, z*=(1-(1.0/p^2))); print("10^"n" ",sqrt(6/z)))

pi estimated from the product of all primes less than 10^n:

10^0 2.4494897427831780981972840747058913920
10^1 3.0935921676911454192536940842087145469
10^2 3.1387371950717209555231899287777680519
10^3 3.1413931847879282123372947308786365411
10^4 3.1415772351603451399115522305263190950
10^5 3.1415913934202453316750296375573376397
10^6 3.1415925471279879791648160810012226847
10^7 3.1415926443848380990491814377585237930
10^8 3.1415926527790129584805149679019884804
10^9 3.1415926535173431626391708456483210697
10^10 3.1415926535832448165410657025850891080

cpu time = 7min, 50,161 ms, real time = 7min, 50,326 ms.

each power of 10 yields slightly more than one more correct digit.  asymptotically, probably just one digit, so this is not an efficient method of calculating pi, though it is nice to have gotten the digits above without too much effort.

previously, estimating e with primes.

[pglmhtip] how to exit visual mode in ex

everyone wants to know the opposite direction, how to exit line-editing "ex mode" and return to visual mode in vi, so it is tough to find information for this direction.

the correct answer is "Q", but vim 8.2 (package vim-runtime) in Ubuntu 22.04 has remapped Q to something completely unrelated ("formatting") so Q is inaccessible.  (because everyone who hates line-editing also wants to configure vi never to accidentally enter "ex mode").  unlike emacs using M-x, there is no way to call a function by its long name if you just want to use it once.  to make Q exit visual mode for the rest of the session: ":unmap Q".  but of course that changes state, so you are not returning exactly to the same "ex" state that you started with: what if you then want to enter a visual mode with Q again having its remapped "formatting" behavior?  I do not know of a way to restore Q to whatever it was before (without knowing what it was before) (unlike emacs, there is no local-set-key, local-unset-key).  this is a mess.

to start "ex" (or "vi") with Q being available again: "ex -u NONE".  or maybe "ex -u NONE -N".  but "-u NONE" disables all initialization files, not just the Q remapping.  (-u NONE also makes vim behave more like original vi, a large set of differences in behavior.  -N modifies -u NONE to keep vim behaving like vim, which is good if you are used to vim not vi.)  I could not find any way to disable just the Q remapping from the command line.  this is a mess.

(if you want Q permanently available, you can create a user initialization file to override the system initialization file.)

the almost correct answer, even with "Q" remapped and inaccessible, is "gQ".  but "Q" and "gQ" are subtly different (in a way I do not understand).  when vim is invoked as "ex", it starts in "Q" mode, not "gQ" (according :h Q), and, as described above, there is no easy way to go back to Q after entering visual the first time.  this is a mess.

Q mode (but not gQ mode) has been removed from neovim (nvim).  if invoked as "ex", neovim presumably starts in gQ mode, but this is not documented anywhere I could find.

[fixtyxpa] walk may repeat the batter

if a baseball batter is awarded a base on balls, let the manager have the option of instead having the next batter ("batter on deck") take first base and the walked batter repeat the at bat.

not sure if it should be a temporary or permanent change in the batting order.

or, any pinch runner instead of batter on deck.

[ujzkawfu] 45 minutes from now, the human race could be extinct

two clocks, one labeled "Time now" and the other set and labeled "45 minutes into the future".  above them, a sign reading "45 minutes from now, the human race could be be extinct."  in between the clocks, a button with a cover over it labeled "LAUNCH".

more fancy: the T+45min clock stops once the launch button is pressed.  the launch button remains depressed -- it does not retract -- once pushed, indicating it has been pressed.  the T-0 clock runs for 45 minutes until its time agrees with the stopped T+45min clock, then it too stops.  the clocks resume and the button resets after 12 hours (for art).

too fancy: graphical display of missiles in flight.

(low earth orbit = 90 minutes, so max 45 minutes for an ICBM to reach anywhere on earth.  extinction could be quicker than that if we strategically position nukes all over the earth, which we do (sigh).  although it will be difficult to directly kill submarine crews, they are doomed to die from starvation or radiation poisoning after salted nuclear weapons.)

[jxtzkavi] magic hexagon base 6

it seems appropriate to present the unique nontrivial magic hexagon with its numbers expressed in base 6.  final number = 31 (base 6) = 19 (base 10).  magic sum = 102 (base 6) = 38 (base 10).

     3  25  30
  31  11   1  15
24   2   5  10  13
  20   4  12  22
    14  21  23

[typdfiqs] Amtrak Southwest Chief schedule change starting late May 2026

according to "Schedules" on https://www.amtrak.com/ , 2026-05-07:

Thursday, May 28, 2026: Train 3 departs CHI 2:25p arrives LAX 7:57a (43h 32m).  this is the last train on the old schedule.

Friday, May 29, 2026: Train 1003 (a special number) departs CHI 1:30p arrives LAX 8:12a (44h 42m)

Saturday, May 30, 2026: Train 3 departs CHI 1:30p arrives LAX 8:12a (44h 42m)

there is no information about this on "Service Alerts and Notices" https://www.amtrak.com/alert

[ynwwppvz] estimating e by counting primes

let n = 2^22 = 4194304.  there are p = primepi(n) = 295947 primes less than n.

using the prime number theorem n/log(n) ~= primepi(n), then e ~= n^(p/n) ~= 2.93 .

using n/(log(n) - 1) ~= primepi(n), then e ~= n^(p/(n+p)) ~= 2.732 .

(the same n was used previously computing e with primorial.  using the values of primes as we did then yielded slightly more precision (2.717) than using just their count as we do now.)

the typical way of computing fractional powers is with log and exp, but if you have such functions, then e = exp(1).

we chose n to be a power of 2 to permit easily computing the fractional power ^(p/n) in the first approximation without needing log and exp: the 1/n power in the can be computed by repeated square roots.  n^(p/n) = (n^p)^(1/n).  compute n^p first by augmenting machine precision floating point with an exponent that can become much larger than provided by the machine type.  the only operations one needs are multiplication and square root.  example implementation in Haskell.  there are some bits (pun!) of symmetry between exponentiation by squaring for the numerator of p/n followed by repeated square root for the denominator.

for the second approximation n^(p/(n+p)), one could have attempted to choose a different n such that n+p is a power of two.  but 3916608 + primepi(3916608) = 2^22-1 and 3916609 + primepi(3916609) = 2^22+1 (because 3916609 is prime), so it would have required some fudging.

better is to approximate the fractional exponent p/(n+p) as round(2^64 * p/(n+p))/2^64 .  (64 bits should be enough for everyone.)  for n = 2^22, p = 295947, this works out to to 1215802539408625636 / 2^64.

or, use Newton's method to compute the (n+p)th root of x^p.

comparing with methods that don't involve primes, but inputs of a similar scale:

e ~= (1+1/n)^n : for n=2^22, e ~= 2.718281504 .

sum(n=0, 10, 1.0/n!) ~= 2.71828180 , choosing 10 because 10! is the largest factorial less than 2^22 .

[esowrowp] speaking leading zeroes

zeroty ten twenty thirty forty fifty sixty seventy eighty ninety

0 zero
1 one
00 zeroty
01 zeroty one
000 zero hundred
001 zero hundred one
0000 zero thousand
0001 zero thousand one
00000 zeroty thousand
00001 zeroty thousand one
01000 zeroty one thousand
000000 zero hundred thousand

or "ohty" instead of zeroty to avoid similarity between zero and zeroty?  but ohty sounds similar to eighty.

or "aughty"?  also sounds similar to eighty.

or "noughty"?  context probably distinguishes it from naughty.

"nohty" ("notey")?

typically digit strings that are not numbers should be read as digit strings, but place value indicators inserted into the string serve as useful signposts.  in this case, also consider speaking internal zeroes: 

000000 zero hundred zeroty zero thousand, zero hundred zeroty zero
20 twenty zero

zip codes

10 "ten" remains irregular.  though could we do something like 10 one-ty zero, 20 two-ty zero (future post knlwtvtb).

[pfbjvubj] environment variables modifying Makefile variables

consider the following GNU Makefile and invocations in the bash shell:

C = c
Q ?= q
.PHONY:default
default:
        @echo C=$(C)
        @echo Q=$(Q)

$ make
C=c
Q=q

$ make C=notc
C=notc
Q=q

$ make Q=notq
C=c
Q=notq

$ C=notc make
C=c
Q=q

$ Q=notq make
C=c
Q=notq

either type of assignment can be overridden by specifying a new value after "make" on the command line, but only variables assigned with ?= can also be overridden by the environment.  I've deliberately done both: makefiles in which make communicates with its caller via environment variables, and makefiles in which make should ignore any identically named environment variables.

Next, experimenting with += :

P += p
.PHONY:default
default:
        @echo P=$(P)

$ make
P=p

$ make P=prefix
P=prefix

$ P=prefix make
P=prefix p

like with ?= , only user-specified values in the environment can be appended to with += .  this is especially useful is the following idiom: if a Makefile specifies "CFLAGS += necessary flags", then the user can add additional flags to CFLAGS just for one build without editing the Makefile.  for += to work keeping the necessary CFLAGS specified in the Makefile, the user must specify CFLAGS in the environment as demonstrated in the last example above, not in a command-line argument to make.

an attacker could surreptitiously set CFLAGS="something evil" in your environment, thereby affecting all builds that use exclusively CFLAGS += , but if you are worried about an attacker who can set and export arbitrary variables in your environment, you probably have bigger problems.

future work: the "override" makefile directive.

[obaxedbg] which half point

rate chess players by how often they turn a drawn position into a lost one, and by how often they turn a winning position into a drawn one.  (turning a winning position into a lost one counts as one of each.)  the best at avoiding bad moves of the first type might not be the best at avoiding the other type.

devilish detail of determining ground truth of a position.  maybe play out by an engine.

devilish detail of normalization: some positions are easier to make mistakes in, and some players probably (as a result of play style) encounter such positions more frequently.

skill difference might matter: a subtle (but still half-point) mistake against a beginner might not matter, because the beginner might not be able to exploit it, or might be expected to blunder it back later in the game.

having a high rate at either or both of these categories might not equate to bad results -- perhaps you have a playing style that, even though you make a lot of mistakes, you induce your opponents to make even more.

having a low rate of making mistakes might not equate to good results: only one mistake per game is enough to lose every game.

opposite measure: how often a player induces an opponent to make each of the half-point mistakes.  similar devilish details, including normalization.

[rgvdheir] random convex polytope

first, create a random smooth shape by picking random weights for the first few spherical harmonics and summing them.  reject if the object is not convex: determining this seems difficult.  can the coefficients be chosen to favor convexity?  previously 2D.

then, uniformly randomly sample some points on the surface and construct tangent planes, forming a random convex polyhedron (might not be closed).  uniform sampling of surface points on an arbitrary surface seems difficult.  maybe a random walk of small steps each of approximately constant geodesic length.  how many steps?

or, permit starting with a smooth shape not necessarily convex.  reject a tangent plane if its inward half space does not contain the origin.

extend this construction to higher dimensions.  what are the hyper-spherical harmonics?  curse of dimensionality probably somehow happens.

a 4D polytope (polychoron) is interesting because its cells are random polyhedra whose faces fit together pairwise but not further in 3D.  how can a collection of specified irregular polyhedra be manufactured?  3D printing is of course one way.  or, mill some faces, but include nubs on one or more faces.  then, reorient the piece upside down, holding it in place from below by the nubs.  mill the remaining faces.  then, somehow remove the nubs.

mill two negative halves then cast then join somehow?

create a net and fold (origami)?

the edges of a random convex 3D polyhedron define a random planar graph.  do analogous graphs for convex polytopes in higher dimensions have a property analogous to planarity?

Delaunay triangulation is another way to get a random planar graph, but faces are generally always triangles.  but are most of the faces produced by the tangent method described above also triangles?

3D convex hull of random points results in a convex polyhedron, and this generalizes to arbitrary dimensions.  I think most faces will again be triangles (simplices).  can the points be sampled in a manner to get interesting shapes?  for example, after the first few points, avoid new points that are way outside the convex hull so far, drastically changing the polyhedron.

aside: the minimal surface in 3D that contains all given points is not necessarily the convex hull as it is in 2D.  the minimal surface might have hyperbolic paraboloid faces.

a random convex polytope generates a random linear or quadratic programming problem.

applying a linear transformation to a convex polytope maintains its convexity.  proof: assume false, meaning that after the linear transformation there is a line segment that starts inside, goes outside, then comes back inside (definition of non-convex).  apply the inverse linear transformation, which is linear.  the line segment remains a line segment.  it still goes in out in.  the existence of such a line segment in the initial polytope means it was not convex.

consider relaxing the requirement of convexity and only require non-self-intersecting.  one can have holes and other weird features, though perhaps some so weird that it might be good to forbid them.  what weird features, especially in higher dimensions?  polygonal faces can be non-convex with holes.  that is probably OK, but faces with intersecting edges (e.g., pentagram) are not OK.

previous thoughts on random polyhedra..

[ibrwpoia] full coverage by uniform sampling

B different balls in an urn.  uniformly draw with replacement N times.  what is the probability that all B balls are seen during the N draws?

given B, for what value of N is the probability (about) 50%?

simulation in Pari/GP:

u(trials, B, N)= my(success=0); for(c=1, trials, my(a=vector(B)); for(i=1, N, r=1+random(B); a[r]=1); good=1; for(i=1, B, if(0==a[i], good=0; break)); success+=good); 1.0*success/trials;

B N
2 2
4 7
8 19-20
10 26-27
16 50
20 67
30 113
32 122
40 162
50 213-214
60 267-268
64 289-290

B=64 balls, N=290 draws:
u(10^5, 64, 290)
0.50303

cpu time = 21,920 ms, real time = 21,924 ms.

exact answer: 1 - probability of failure by the inclusion-exclusion principle, simplifies to:

v(B, N)= sum(i=0, B, (-1)^i * (B-i)^N * binomial(B, i)) /B^N

1.*v(64,290)
0.504257282

exact bounds:

10 [26, 27]
20 [66, 67]
30 [112, 113]
40 [161, 162]
50 [213, 214]
60 [267, 268]
70 [322, 323]
80 [379, 380]
90 [437, 438]
100 [496, 497]
110 [556, 557]
120 [618, 619]
130 [679, 680]
140 [742, 743]
150 [806, 807]
160 [870, 871]
170 [934, 935]
180 [1000, 1001]
190 [1066, 1067]
200 [1132, 1133]
210 [1199, 1200]
220 [1266, 1267]
230 [1334, 1335]
240 [1402, 1403]
250 [1471, 1472]
260 [1540, 1541]

2 [2, 2]
4 [6, 7]
8 [19, 20]
16 [50, 51]
32 [122, 123]
64 [289, 290]
128 [667, 668]
256 [1512, 1513]
512 [3381, 3382]
1024 [7472, 7473]
2048 [16364, 16365]
4096 [35569, 35570]

the growth rate of N seems to be O(B*log B).  let's use N = B*log B exactly and calculate the probability of full coverage:

v(B, round(B*log(B)))

B N v
4 6 0.380859375
8 17 0.365562211
16 44 0.347014424
32 111 0.363466498
64 266 0.364065310
128 621 0.366295109
256 1420 0.367804277
512 3194 0.367507854
1024 7098 0.367763404
2048 15615 0.367750482
4096 34070 0.367873303

the probability seems to be converging to exp(-1) = 0.367879441 .

how many draws N are necessary to achieve at least a given probability of full coverage?

previously similar: Euler's Game of Coincidence

[fieutgtv] sum of the reciprocals of known primes

at the time of writing, the largest known prime number is the Mersenne prime 2^136279841 - 1, a 41-million-digit number.

the sum of the reciprocals of all primes up to the largest known prime can be estimated to be approximately

log(log(2^136279841 - 1)) + 0.261497 ~= 18.6252

, where the offset 0.261497 is the Meissel-Mertens constant (known to many more digits).  (how many more?)

(previously, on log(2^n +1).)

however, the sum of the reciprocals of all primes up to infinity is infinity, i.e., the infinite sum diverges (proven by Euler).  despite our finite sum being up to a huge prime, 18.6 is still a long way from infinity.

the reciprocal of the next prime after the largest known prime ("the first unknown prime") will add approximately 10^-41000000 to our partial sum 18.6252 , a tiny step.  and each additional step will be yet smaller, because reciprocals keep getting smaller.  yet infinity will still eventually be "reached": the sum will exceed any finite number you can think of: Graham's number, Rayo's number, Loader's number, etc., (and all finite numbers you can't think of).

previously, computing the sum one button press at a time.

[wlxarken] hope nand comfort

hypothesis: a society may grant its lower classes hope for upward social mobility, or a comfortable existence, but not both.

this is a depressing hypothesis similar in spirit to those that originally got economics labeled the dismal science.

the general ideas justifying the hypothesis are the following.  if you grant the lower classes the possibility of upward social mobility, only some of them will be permitted to achieve it, so they will need to compete against each other in a zero-sum game for who achieves it.  zero-sum games are cutthroat, not comfortable.  if you grant (enough) comfort, there won't be incentive to seek upward social mobility (this might require more thought).  if you grant comfortable upward social mobility, then the lower classes will become depopulated, so society won't be able to function because it needs lower classes to do lower-class stuff.

also seemingly in support of this hypothesis are highly unequal, highly stratified societies which have remained stable for long periods of time.  they presumably deny upward mobility but grant enough relative comfort to squelch social upheaval.

unfortunately, or fortunately, this hypothesis is only a vague idea which could be incorrect in many ways.

Solow famously disproved Malthus's original "dismal science" with the economic concept of technology.  however, our dismal hypothesis survives overall positive sum: technology may lift the entire pyramid, but it still remains a pyramid.

society can grant its lower classes neither hope nor comfort, which is why the conjunction is nand not xor.  but a situation with neither hope nor comfort might be unstable, threatening civil war.

[utglyvmo] dark matter cloud

what would we observe if the earth, or the solar system, passed though a dense cloud of dark matter?

assume WIMP.  we would see not the dark matter directly but only indirect effects.  hypothesize that orbits will change.  will the sun gravitationally capture dark matter and increase in mass, increasing its fusion rate?

do dense clouds of dark matter exist?  how dense might they be?

update: a low-surface-brightness galaxy could be considered a galaxy-sized dense cloud of dark matter.  but I don't think their average dark matter density is that much higher than that of the Milky Way.

previously on dark matter in the solar system.

[sebqohhz] Star Wars by ancient Greek playwrights

pretend that the spirits of ancient Greek playwrights had possessed George Lucas and teamed up within his mind to write the actual screenplay to Star Wars.  who wrote which part?

Aeschylus:

[opening shot: stars.  small spaceship.  laser fire.  huge Imperial Star Destroyer.]

also:

Obi-Wan Kenobi: The Force is what gives a Jedi his power.  It's an energy field created by all living things.  It surrounds us and penetrates us.  It binds the galaxy together.

Sophocles:

Ben Kenobi: Yes, I was once a Jedi knight, the same as your father.
Luke Skywalker: I wish I'd known him.
Ben Kenobi: He was the best star pilot in the galaxy, and a cunning warrior.  I understand that you've become quite a good pilot yourself.
[sorrowfully]
Ben Kenobi: And he was a good friend.

Euripides:

Luke: But I was going into Tosche Station to pick up some power converters!

Aristophanes: Robot Chicken

[pmycapmx] horse meat

cow = beef
pig = pork
sheep = mutton
deer = venison
horse = ?

we have alternate names for the meat of animals that we maybe do not want to think of as animals when we eat them.  Americans and British generally do not like to eat horse.  would we be more willing if there were an alternate English name for horse meat?  in cultures which do eat horse, do they have an alternate name for its meat (for any meat)?  other animals (e.g., dog)?

in English, chicken = chicken, fish = fish, but these we consider "lower" animals.  however, lamb = lamb despite sheep = mutton.

maybe horse will become demoted to lower animal soon because of the ascendancy of the automobile.  will horse-riding as a hobby of the upper class affect its transition to acceptability as food?  how is it currently affecting?  on one hand, horses are (expensive) friends not food to those who ride them.  on the other hand, horses are literally below them when ridden.

[ygnpgfnt] potential energy of a compressible ball

model a spherical solid as a lattice of atoms connected by springs.  let the solid's own gravity compress itself.  how much energy gets stored in the compressed springs?

the earth's nickel-iron core is denser than nickel-iron at room temperature, denser than nickel-iron meteorites.

aside: does this mean that destroying a planet requires less than its gravitational binding energy?  harness this internal potential energy.  but this requires more thought: gravitational potential energy balances spring energy.

to model the lattice of springs, we need spring potential energy to approach infinity as distance between atoms approaches zero.  Hooke's Law is not the right model for this.

aside: does it matter what lattice (e.g., cubic, rhombic dodecahedral) of springs?

we also need potential energy to head to infinity as distance increases to infinity so that atoms do not convert to gas and evaporate in our model.  Hooke's Law could work for this direction.

U = x^2/2 + 1/x.  first term is Hooke's Law; second is electrostatic repulsion.  the leading coefficient of 1/2 places the local minimum at x=1.

alternatively, a U-shaped function that has local minimum at x=1 and is infinite at x=0 and x=2: U = 1/x - 1/(x-2) = 2/(x*(2-x)).  the two expressions after U= are the two most straightforward ways of constructing a function with the two vertical asymptotes.  it is neat that they yield the same result.

consider general relativity: the potential energy of compressed strings exerts gravity, bends spacetime.

[efggjrfp] infinity breaking groups

consider adding one infinite element to the integers or reals under addition and seeing if it remains a group.

first, consider defining 1 + infinity = infinity in the style of Cantor's cardinal numbers.

(1 + infinity) + (-infinity)
= infinity + (-infinity)
= 0

1 + (infinity + (-infinity))
= 1 + 0
= 1

associativity violated, so not a group.  note that, by group axioms, -infinity must exist, and -infinity + infinity = 0.

ordinal numbers are not a group because omega + (-1) is not defined.

the Riemann sphere (complex numbers extended with infinity) is not a group because infinity + (-infinity) is not defined.  (-infinity) itself might not be defined.

some ways to extend reals with infinity to get a group: Levi-Civita field (not too difficult to understand), surreal numbers (rather difficult to understand).  yet more exist, typically also difficult to understand.

the above successful ways get you not just a group but a field (with multiplication and division).  is that necessary?

unstated: why do you even want a group of numbers with infinity?

[sguflopg] prime(x) = N*x

the 6455th prime number is 64553, a factor of about 10.

here are a few variations on estimating the crossover point using the prime number theorem.

if approximating the nth prime by n*log(n) :

nthprime(i) = 10*i
i*log(i) = 10*i
log(i) = 10
i = exp(10)
i ~= 22026.5

crossover predicted around the 22026th or 22027th prime, namely 249779 or 249797.

if approximating primepi(n)= n/log(n) :

primepi(n) = n/10
n/log(n) = n/10
log(n) = 10
n = exp(10)
n ~= 22026.5

crossover at 22026.5 (bracketed by primes 22013 and 22027 which have indices 2466 and 2467 respectively).

if approximating primepi(n)= n/(log(n)-1) :

primepi(n) = n/10
n/(log(n)-1) = n/10
solve(n=1000, 100000, n/(log(n)-1) - n/10)
n ~= 59874.1417

crossover predicted around 59874.1 which is bracketed by primes 59863 and 59879 which have indices 6048 and 6049 respectively.  this is the best approximation, albeit not closed form.

because exp grows exponentially (like it says on the tin), we will never know the exact crossover for larger factors, e.g., 100.

enumerate the crossover points for small integer factors.  there is much sophistication available for counting primes without explicitly generating them.

as of 2022, the record evaluation of the prime counting function is primepi(10^29) ~= primepi(exp(66.8)) ~= 10^29/65.8 . it will take a little more work to find the crossover point nthprime(x) = 65*x.